apdx_invariants.tex

For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator.
There is generally no strict conservation of mass,
as the equation of state is non linear with respect to $T$ and $S$.
In practice, the mass is conserved to a very high accuracy.
%% =================================================================================================
\subsection{Advection term}
\label{subsec:INVARIANTS_5.1}

conservation of a tracer, $T$:
\[
  \frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right)
  =  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0
\]

conservation of its variance:
\begin{flalign*}
  \frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right)
  =&  \int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }
  -   \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }
\end{flalign*}

Whatever the advection scheme considered it conserves of the tracer content as
all the scheme are written in flux form.
Indeed, let $T$ be the tracer and its $\tau_u$, $\tau_v$, and $\tau_w$ interpolated values at velocity point
(whatever the interpolation is),
the conservation of the tracer content due to the advection tendency is obtained as follows:
\begin{flalign*}
  &\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\
  &\equiv - \sum\limits_{i,j,k}    \biggl\{
  \frac{1} {b_t}  \left(  \delta_i    \left[   U \;\tau_u   \right]
    + \delta_j    \left[   V  \;\tau_v   \right] \right)
  + \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right]    \biggl\}  b_t   &&&\\
  %
  &\equiv - \sum\limits_{i,j,k}     \left\{
    \delta_i  \left[   U \;\tau_u   \right]
    + \delta_j  \left[   V  \;\tau_v   \right]
 	 + \delta_k \left[ W \;\tau_w \right] \right\}   && \\
  &\equiv 0 &&&
\end{flalign*}

The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme,
\ie\ when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$.
It can be demonstarted as follows:
\begin{flalign*}
  &\int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }
  = - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
  \equiv& - \sum\limits_{i,j,k} T\;
  \left\{
    \delta_i  \left[ U  \overline T^{\,i+1/2}  \right]
    + \delta_j  \left[ V  \overline T^{\,j+1/2}  \right]
    + \delta_k \left[ W \overline T^{\,k+1/2} \right]          \right\} && \\
  \equiv& + \sum\limits_{i,j,k}
  \left\{     U  \overline T^{\,i+1/2} \,\delta_{i+1/2}  \left[ T \right]
    +  V  \overline T^{\,j+1/2} \;\delta_{j+1/2}  \left[ T \right]
    +  W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right]     \right\}      &&\\
  \equiv&  + \frac{1} {2}  \sum\limits_{i,j,k}
  \Bigl\{   U  \;\delta_{i+1/2} \left[ T^2 \right]
  + V  \;\delta_{j+1/2}  \left[ T^2 \right]
  + W \;\delta_{k+1/2} \left[ T^2 \right]   \Bigr\}     && \\
  \equiv& - \frac{1} {2}  \sum\limits_{i,j,k} T^2
  \Bigl\{    \delta_i  \left[ U  \right]
  + \delta_j  \left[ V  \right]
  + \delta_k \left[ W \right]     \Bigr\}      &&&  \\
  \equiv& + \frac{1} {2}  \sum\limits_{i,j,k} T^2
  \Bigl\{   \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t}     \Bigr\}      &&& \\
\end{flalign*}
which is the discrete form of $ \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }$.

%% =================================================================================================
\section{Conservation properties on lateral momentum physics}
\label{sec:INVARIANTS_dynldf_properties}

The discrete formulation of the horizontal diffusion of momentum ensures
the conservation of potential vorticity and the horizontal divergence,
and the dissipation of the square of these quantities
(\ie\ enstrophy and the variance of the horizontal divergence) as well as
the dissipation of the horizontal kinetic energy.
In particular, when the eddy coefficients are horizontally uniform,
it ensures a complete separation of vorticity and horizontal divergence fields,
so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence
(variance of the horizontal divergence) and \textit{vice versa}.

These properties of the horizontal diffusion operator are a direct consequence of
properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}.
When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken,
the term associated with the horizontal gradient of the divergence is locally zero.

%% =================================================================================================
\subsection{Conservation of potential vorticity}
\label{subsec:INVARIANTS_6.1}

The lateral momentum diffusion term conserves the potential vorticity:
\begin{flalign*}
  &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
  \Bigl[    \nabla_h  \left( A^{\,lm}\;\chi  \right)
  - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)    \Bigr]\;dv   \\
  % \end{flalign*}
  %%%%%%%%%% recheck here....  (gm)
  % \begin{flalign*}
  =& \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times
  \Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)  \Bigr]\;dv  \\
  % \end{flalign*}
  % \begin{flalign*}
  \equiv& \sum\limits_{i,j}
  \left\{
    \delta_{i+1/2} \left[  \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
    + \delta_{j+1/2} \left[  \frac {e_{1u}} {e_{2u}\,e_{3u}}  \delta_j \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
  \right\} 	 \\
  %
  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
  %
  \equiv& \sum\limits_{i,j,k}
  -\,\left\{
    \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i	\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right]
	 + \frac{e_{1u}} {e_{2u}\,e_{3u}}  \delta_j 	\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_j \left[ 1\right]
  \right\} \quad \equiv 0
  \\
\end{flalign*}

%% =================================================================================================
\subsection{Dissipation of horizontal kinetic energy}
\label{subsec:INVARIANTS_6.2}

The lateral momentum diffusion term dissipates the horizontal kinetic energy:
%\begin{flalign*}
\[
  \begin{split}
    \int_D \textbf{U}_h \cdot
    \left[ \nabla_h 		\right.   &     \left.       \left( A^{\,lm}\;\chi \right)
      - \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\
    \\  %%%
    \equiv& \sum\limits_{i,j,k}
    \left\{
      \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right]
      - \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right]
    \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\
    &\;\; + 	\left\{
	   \frac{1} {e_{2u}}             \delta_{j+1/2}	\left[ A_T^{\,lm}          \chi    \right]
      + \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
    \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\
    \\  %%%
    \equiv& \sum\limits_{i,j,k}
    \Bigl\{
    e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right]
    - e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right]
	 \Bigl\}
	 \\
    &\;\; + \Bigl\{
    e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right]
    + e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
    \Bigl\}      \\
    \\  %%%
    \equiv& \sum\limits_{i,j,k}
    - \Bigl(
    \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right]
    + \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right]
    \Bigr) \;  A_T^{\,lm} \chi   \\
    &\;\; - \Bigl(
    \delta_{i+1/2}  \left[  e_{2v}  \;v  \right]
    - \delta_{j+1/2}  \left[  e_{1u} \;u  \right]
    \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\
    \\  %%%
    \equiv& \sum\limits_{i,j,k}
    - A_T^{\,lm}  \,\chi^2   \;e_{1t}\,e_{2t}\,e_{3t}
    - A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}
    \quad \leq 0       \\
  \end{split}
\]

%% =================================================================================================
\subsection{Dissipation of enstrophy}
\label{subsec:INVARIANTS_6.3}

The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform:
\begin{flalign*}
  &\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times
  \left[   \nabla_h \left( A^{\,lm}\;\chi  \right)
    - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)   \right]\;dv &&&\\
  &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times
  \left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\
  &\quad \equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f}
  \left\{     \delta_{i+1/2} \left[  \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta  \right]   \right]
    + \delta_{j+1/2} \left[  \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta  \right]   \right]      \right\}   &&&\\
  %
  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
  %
  &\quad \equiv  - A^{\,lm} \sum\limits_{i,j,k}
  \left\{    \left(  \frac{1} {e_{1v}\,e_{3v}}  \delta_i \left[ e_{3f} \zeta  \right]  \right)^2   b_v
    + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}  \quad \leq \;0    &&&\\
\end{flalign*}

%% =================================================================================================
\subsection{Conservation of horizontal divergence}
\label{subsec:INVARIANTS_6.4}

When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken,
the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}.
The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform.
\begin{flalign*}
  & \int\limits_D  \nabla_h \cdot
  \Bigl[     \nabla_h \left( A^{\,lm}\;\chi \right)
  - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \Bigr]  dv
  = \int\limits_D  \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi  \right)   dv   \\
  %
  &\equiv \sum\limits_{i,j,k}
  \left\{   \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]  \right]
    + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} \left[ \chi \right]  \right]    \right\}    \\
  %
  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
  %
  &\equiv \sum\limits_{i,j,k}
  - \left\{   \frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right]
    + \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right]    \right\}
  \quad \equiv 0
\end{flalign*}

%% =================================================================================================
\subsection{Dissipation of horizontal divergence variance}
\label{subsec:INVARIANTS_6.5}

\begin{flalign*}
  &\int\limits_D \chi \;\nabla_h \cdot
  \left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right)
    - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv
  = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    \\
  %
  &\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1t}\,e_{2t}\,e_{3t}}  \chi
  \left\{
    \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right]
    + \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right]
  \right\} \;   e_{1t}\,e_{2t}\,e_{3t}    \\
  %
  \intertext{Using \autoref{eq:DOM_di_adj}, it turns out to be:}
  %
  &\equiv - A^{\,lm} \sum\limits_{i,j,k}
  \left\{    \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  b_u
    + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\}
  \quad \leq 0
\end{flalign*}

%% =================================================================================================
\section{Conservation properties on vertical momentum physics}
\label{sec:INVARIANTS_7}

As for the lateral momentum physics,
the continuous form of the vertical diffusion of momentum satisfies several integral constraints.
The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy:
\begin{align*}
  \int\limits_D   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
  \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;  dv
  \qquad \quad &= \vec{\textbf{0}}
  %
  \intertext{and}
  %
                 \int\limits_D
                 \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
                 \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\; dv    \quad &\leq 0
\end{align*}

The first property is obvious.
The second results from:
\begin{flalign*}
  \int\limits_D
  \textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
  \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;dv    &&&\\
\end{flalign*}
\begin{flalign*}
  &\equiv \sum\limits_{i,j,k}
  \left(
    u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2}  \left[ u \right]  \right]\;  e_{1u}\,e_{2u}
    + v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}   \left[ v \right]  \right]\;  e_{1v}\,e_{2v} \right)   &&&
  %
  \intertext{since the horizontal scale factor does not depend on $k$, it follows:}
  %
  &\equiv - \sum\limits_{i,j,k}
  \left(  \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u}
    + \frac{A_v^{\,vm}} {e_{3vw}}  \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v}  \right)
  \quad \leq 0   &&&
\end{flalign*}

The vorticity is also conserved.
Indeed:
\begin{flalign*}
  \int \limits_D
  \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}  \left(
      \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k}
    \right)  \right)\; dv   &&&
\end{flalign*}
\begin{flalign*}
  \equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
  \bigg\{    \biggr.   \quad
  \delta_{i+1/2}
  &\left(   \frac{e_{2v}} {e_{3v}} \delta_k  \left[  \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right]  \right]  \right)   &&\\
  \biggl.
  - \delta_{j+1/2}
  &\left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[  \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right]  \right]   \right)
  \biggr\} \;
  e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   &&
\end{flalign*}

If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, \ie
\begin{flalign*}
  \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times
  \left(   \frac{1} {e_3}\; \frac{\partial } {\partial k}
    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)   \right)\; dv = 0   &&&
\end{flalign*}

This property is only satisfied in $z$-coordinates:
\begin{flalign*}
  \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times
  \left(  \frac{1} {e_3}\; \frac{\partial } {\partial k}
    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right)   \right)\; dv   &&&
\end{flalign*}
\begin{flalign*}
  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
  \biggl\{ 	\biggr.	\quad
  \delta_{i+1/2}
  &\left(   \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v]  \right]   \right)   &&\\
  - \delta_{j+1/2}
  &\biggl.
  \left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u]  \right]    \right) \biggr\}   &&
\end{flalign*}
\begin{flalign*}
  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f}
  \biggl\{		\biggr.	\quad
  \frac{1} {e_{3v}} \delta_k
  &\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right]   \right]    &&\\
  \biggl.
  - \frac{1} {e_{3u}} \delta_k
  &\left[  \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right]  \right]  \biggr\}  &&
\end{flalign*}
Using the fact that the vertical diffusion coefficients are uniform,
and that in $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so that:
$e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows:
\begin{flalign*}
  \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k
  \left[   \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[   \delta_{i+1/2} \left[ e_{2v}\,v \right]
    - \delta_{j+1/ 2} \left[ e_{1u}\,u \right]   \Bigr]    \right]    &&&
\end{flalign*}
\begin{flalign*}
  \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}
  \left( \delta_{k+1/2} \left[ \zeta  \right] \right)^2 \; e_{1f}\,e_{2f}  \; \leq 0    &&&
\end{flalign*}
Similarly, the horizontal divergence is obviously conserved:

\begin{flalign*}
  \int\limits_D \nabla \cdot
  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0    &&&
\end{flalign*}
and the square of the horizontal divergence decreases (\ie\ the horizontal divergence is dissipated) if
the vertical diffusion coefficient is uniform over the whole domain:

\begin{flalign*}
  \int\limits_D \chi \;\nabla \cdot
  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right) \right)\;  dv = 0  &&&
\end{flalign*}
This property is only satisfied in the $z$-coordinate:
\begin{flalign*}
  \int\limits_D \chi \;\nabla \cdot
  \left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
    \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)  \right)\; dv    &&&
\end{flalign*}
\begin{flalign*}
  \equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}}
  \biggl\{ 	\Biggr.	\quad
  \delta_{i+1/2}
  &\left(   \frac{e_{2u}} {e_{3u}} \delta_k
    \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right)    &&\\
  \Biggl.
  + \delta_{j+1/2}
  &\left( \frac{e_{1v}} {e_{3v}} \delta_k
    \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right]   \right)
  \Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   &&
\end{flalign*}

\begin{flalign*}
  \equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \,
  \biggl\{	\biggr.	\quad
  \delta_{i+1/2}
  &\left(
    \delta_k \left[
      \frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right]   \right)    && \\
  \biggl.
  + \delta_{j+1/2}
  &\left(    \delta_k \left[
      \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right]   \right)   \biggr\}    &&
\end{flalign*}

\begin{flalign*}
  \equiv -A^{\,vm} \sum\limits_{i,j,k}
  \frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{
  \delta_{k+1/2} \Bigl[
  \delta_{i+1/2} \left[ e_{2u}\,u \right]
  + \delta_{j+1/2} \left[ e_{1v}\,v \right]  \Bigr]    \biggr\}    &&&
\end{flalign*}

\begin{flalign*}
  \equiv -A^{\,vm} \sum\limits_{i,j,k}
  \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right]   &&&
\end{flalign*}

\begin{flalign*}
  \equiv -A^{\,vm} \sum\limits_{i,j,k}
  \frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right]  \right)^2     \quad  \equiv 0    &&&
\end{flalign*}

%% =================================================================================================
\section{Conservation properties on tracer physics}
\label{sec:INVARIANTS_8}

The numerical schemes used for tracer subgridscale physics are written such that
the heat and salt contents are conserved (equations in flux form).
Since a flux form is used to compute the temperature and salinity,
the quadratic form of these quantities (\ie\ their variance) globally tends to diminish.
As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear.

%% =================================================================================================
\subsection{Conservation of tracers}
\label{subsec:INVARIANTS_8.1}

constraint of conservation of tracers:
\begin{flalign*}
  &\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&& \\ \\
  &\equiv \sum\limits_{i,j,k}
  \biggl\{ 	\biggr.
  \delta_i
  \left[
    A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2}
    \left[ T \right]
  \right]
  + \delta_j
  \left[
    A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2}
    \left[ T \right]
  \right] && \\
  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
  + \delta_k
  \left[
    A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2}
    \left[ T \right]
  \right]
  \biggr\}   \quad  \equiv 0
  &&
\end{flalign*}

In fact, this property simply results from the flux form of the operator.

%% =================================================================================================
\subsection{Dissipation of tracer variance}
\label{subsec:INVARIANTS_8.2}

constraint on the dissipation of tracer variance:
\begin{flalign*}
  \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv	&&&\\
  &\equiv   \sum\limits_{i,j,k} \; T
  \biggl\{  \biggr.
  \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
  & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right]
  \quad&& \\
  \biggl.
  &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right]
  \biggr\} &&
\end{flalign*}
\begin{flalign*}
  \equiv - \sum\limits_{i,j,k}
  \biggl\{ 	\biggr.	\quad
  &    A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right]  \right)^2   e_{1u}\,e_{2u}\,e_{3u}    && \\
  & + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right]  \right)^2   e_{1v}\,e_{2v}\,e_{3v}     && \\ \biggl.
  & + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right]   \right)^2    e_{1w}\,e_{2w}\,e_{3w}   \biggr\}
  \quad      \leq 0      &&
\end{flalign*}

%%%%  end of appendix in gm comment
%}

\subinc{\input{../../global/epilogue}}

\end{document}